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\n a:=c; \n while not a=g[3] do \n v:=[op( v),a]; \n a:=g[2](a,c) \n od; \n v\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "isCP:=proc(g,m) \n loca l i,j;\n i:=1; \n while (i<=nops(g) and \{seq(m(g[i],g [j]),j=1..nops(g))\}=\{op(g)\}) do \n i:=i+1 \n o d; \n evalb(i=nops(g)+1) \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "isAssociative:=proc(g,m) \n local i,j,k;\n i:= 1; \n j:=1; \n k:=1; \n while(i<=nops(g) and m(m(g[i],g[ j]),g[k])=m(g[i],m(g[j],g[k]))) do \n if k=nops(g) then \n \+ if j=nops(g) then \n i:=i+1; \n \+ j:=1; \n k:= 1 \n else \n \+ j:=j+1 \n fi \n else \n \+ k:=k+1 \n fi od; \n evalb(i=nops(g)+1) \nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "hasRightId:=proc(g,m) \n \+ local i,k; \n member(g[1],[seq(m(g[1],g[i]), i=1..nops(g))],'k'); \+ \n i:=1;\n while (i<=nops(g) and m(g[i],g[k])=g[i]) do \n \+ i:=i+1 \n od; \n evalb(i=nops(g)+1) \nend: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "`type/group`:=proc(g) \ntype(g,lis t) and nops(g)=2 and (type(g[1],list) or type(g[1],set)) and type(g[2] ,procedure) and isCP(op(g)) and isAssociative(op(g)) and hasRightId(op (g)) end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "`type/extend edGroup`:=proc(g) local i;\nif type(g,list) and nops(g)=4 and type([g[ 1],g[2]],group) and g[2](g[1][1],g[3])=g[1][1] \nthen i:=1; while not \+ i=nops(g[1])+1 and g[2](g[1][i],g[4](g[1][i]))=g[3] do i:=i+1 od; \nev alb(i=nops(g[1])+1) else false fi end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "invU:=proc(a,n) \n local v; \n igcdex(a,n,'v'); \+ \n v mod n \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "invS L:=(A,n)->[[A[2,2],-A[1,2] mod n],[-A[2,1] mod n, A[1,1]]]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "invGL:=proc(A,n) \n local d; \n d:=invU(A[1,1]*A[2,2]-A[1,2]*A[2,1],n);\n [[A[2,2]*d mod \+ n,-A[1,2]*d mod n],[-A[2,1]*d mod n, A[1,1]*d mod n]] \nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "cyclic:=n->[seq([$i..n,$1..i -1],i=1..n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "dihedral:= n->[op(cyclic(n)),seq([n+1-j$j=i..n,n+1-j$j=1..i-1],i=1..n)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "mult:=proc(a,b,g) \n loca l i,v; \n member([seq(g[a][g[b][i]],i=1..nops(g[a]))],g,'v');\n \+ v \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "Cyclic:=n->[[$ 1..n], (a,b)->`if`(a+b-1<=n,a+b-1,(a+b-1)-n), 1, a->`if`(a=1,1,n-a+2)] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "Dihedral:=n->[[$1..2*n ], (a,b)->mult(a,b,dihedral(n)), 1, a->`if`(a=1 or a>n,a,n-a+2)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Un:=n->[un(n),(a,b)->a*b mod n, 1, a->invU(a,n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "m m:=(A,B,n)->[[A[1,1]*B[1,1] +A[1,2]*B[2,1] mod n,A[1,1]*B[1,2] +A[1,2] *B[2,2] mod n ],[A[2,1]*B[1,1] +A[2,2]*B[2,1] mod n,A[2,1]*B[1,2] +A[2 ,2]*B[2,2] mod n]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "GL2: =n->[gl2(n),(a,b)->mm(a,b,n),[[1,0],[0,1]],a->invGL(a,n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "sl2:=n->select(A->evalb(A[1,1]*A[2 ,2]-A[1,2]*A[2,1] mod n=1),\n[seq(seq(seq(seq([[j,i],[l,k]],l=0..n-1), k=0..n-1),j=0..n-1),i=0..n-1)]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "SL2:=n->[sl2(n),(a,b)->mm(a,b,n),[[1,0],[0,1]],a->inv SL(a,n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Z:=n->[[$0..n- 1],(a,b)->a+b mod n, 0, a->-a mod n]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "Ord:=proc(c,g) \n local a,n;\n a:=c; \n n: =1; \n while not a=g[3] do \n n:=n+1; \n a:=g[2]( a,c) \n od; \n n \n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "Nordlist:=proc(d,g) \n local i, \n v; \n v:=[]; \n for i from 1 to nops(g[1]) do \n if Ord(g[1][i],g)=d \+ then \n v:=[op(v),g[1][i]] \n fi od; \n v\n en d:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 205 "CentralizerS:=proc(S ,G) local i,j,v; \nv:=[]; for i to nops(G[1]) do j:=1; while not j=nop s(S)+1 and \nG[2](G[1][i],S[j])=G[2](S[j],G[1][i]) do j:=j+1 od; if j= nops(S)+1 then v:=[op(v),G[1][i]] fi od; v end:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "Center:=G->CentralizerS(G[1],G):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 37 "Subgroups Generated by a Few Elements" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 30 "We already have the procedure " }{TEXT 256 6 "cycleU" } {TEXT -1 46 " listing the elements of a cyclic subgroup of " }{TEXT 257 1 "U" }{TEXT -1 1 "(" }{TEXT 258 1 "n" }{TEXT -1 61 "). Here is th e procedure finding elements of the subgroup of " }{TEXT 259 1 "U" } {TEXT -1 1 "(" }{TEXT 261 1 "n" }{TEXT -1 21 "), generated by a set" } {TEXT 262 2 " s" }{TEXT -1 16 " of elements of " }{TEXT 260 1 "U" } {TEXT -1 1 "(" }{TEXT 263 1 "n" }{TEXT -1 3 "): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 276 "genU:=proc(s,n) \n local i,j,v,vs; \n \+ v:=\{1\}; \n vs:=\{op(s)\} union \{1\}; \n while not v=vs do \+ \n v:=vs; \n for i from 1 to nops(v) do \n for j from 1 to nops(s) do \n vs:=vs union \{v[i]*s[j] mod \+ n\} od \n od \n od; \n v \nend: " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 37 "For example, here is the subgroup of " }{TEXT 287 1 "U " }{TEXT -1 26 "(48) generated by 5 and 7:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "genU(\{5,7\},48);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<*\"\"\"\"\"&\"\"(\"#6\"#D\"#H\"#J\"#N" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "un(48);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#72\"\"\"\" \"&\"\"(\"#6\"#8\"#<\"#>\"#B\"#D\"#H\"#J\"#N\"#P\"#T\"#V\"#Z" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "For a comparison, here are the cyc lic subgroups generated by 5 and 7: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "cycleU(5,48);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\" \"\"\"\"&\"#D\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "cycleU (7,48);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"\"\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Here is the analogous procedure for exten ded groups: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "Gen:=proc( s,g) \n local i,j,v,vs; \n v:=\{g[3]\}; \n vs:=\{op(s)\} union v; \+ \n while not v=vs do \n v:=vs; \n for i from 1 to nops(v) d o \n for j from 1 to nops(s) do \n vs:=vs union \{g[2](v[i],s[j]), g[2](s[j],v[i])\} \n od\n od \n od ; \nv \nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "Z[n];" "6#&%\"ZG6#%\"nG" }{TEXT -1 76 ",we don't actually need s uch a procedure, because the subgroup generated by " }{TEXT 267 4 "s = " }{TEXT -1 1 "\{" }{TEXT 264 12 "a, b, ..., c" }{TEXT -1 41 "\} is j ust a cyclic subgroup generated by " }{TEXT 265 4 "igcd" }{TEXT -1 0 " " }{TEXT 266 9 "( op(s) )" }{TEXT -1 30 ". Let's see what we get using " }{TEXT 268 3 "Gen" }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Gen(\{4,6\},Z(12));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#<(\"\"!\"\"#\"\"%\"\"'\"\")\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The same answer! Now, something a little bit more complicated, a s ubgroup of " }{TEXT 270 2 "SL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[3] ;" "6#&%\"ZG6#\"\"$" }{TEXT -1 15 ") generated by " }{XPPEDIT 18 0 "ma trix([[1, 1], [0, 1]]);" "6#-%'matrixG6#7$7$\"\"\"F(7$\"\"!F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "matrix([[0, 1], [2, 0]]);" "6#-%'matrixG6 #7$7$\"\"!\"\"\"7$\"\"#F(" }{TEXT -1 3 " : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "map(matrix,Gen(\{[[1,1],[0,1]],[[0,1],[2,0]]\},GL2 (3)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<:-%'matrixG6#7$7$\"\"!\"\"# 7$\"\"\"F,-F%6#7$F+7$F*F)-F%6#7$7$F)F,7$F*F*-F%6#7$F07$F,F*-F%6#7$F0F5 -F%6#7$7$F*F,F+-F%6#7$F+F4-F%6#7$F4F0-F%6#7$F@F0-F%6#7$F9F5-F%6#7$F57$ F,F)-F%6#7$F@F(-F%6#7$F(F9-F%6#7$F9FP-F%6#7$F9F4-F%6#7$FPF4-F%6#7$F(FP -F%6#7$F5F@-F%6#7$F4F@-F%6#7$F0F(-F%6#7$FPF+-F%6#7$FPF@-F%6#7$F+F9-F%6 #7$F5F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "It looks like " } {TEXT 269 2 "SL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[3];" "6#&%\"ZG6# \"\"$" }{TEXT -1 23 "). Let's check it out: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "evalb(Gen(\{[[1,1],[0,1]],[[0,1],[2,0]]\},GL2(3) )=\{op(sl2(3))\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 "Intersections and Products of Gro ups" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 154 "Some exercises in Dr. Gall ian's text are using intersections of subgroups and a product of group s. The intersections can be found using the Maple command " }{TEXT 271 9 "intersect" }{TEXT -1 48 ". For example, the intersection of sub groups of " }{TEXT 272 1 "U" }{TEXT -1 36 "(48) generated by \{5,7\} a nd \{9,25\}: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "genU(\{5,7 \},48) intersect genU(\{9,25\},48);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<$\"\"\"\"#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Another example, the intersection of cyclic subgroups of " }{XPPEDIT 18 0 "Z[120];" "6 #&%\"ZG6#\"$?\"" }{TEXT -1 30 " generated by 4, 6, and 25. : " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`intersect`(\{op(Cycle(4, Z( 120)))\}, \{op(Cycle(6, Z(120)))\}, \{op(Cycle(25, Z(120)))\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$\"\"!\"#g" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 34 "It must be the cyclic subgroup of " }{XPPEDIT 18 0 "Z[1 20];" "6#&%\"ZG6#\"$?\"" }{TEXT -1 65 " generated by the least common \+ multiple of 4, 6, and 25 mod 120: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Cycle(ilcm(4,6,25) mod 120, Z(120));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!\"#g" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Now, define the product of the extended groups using the following procedure: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "`&x`:=proc (g,h) local m,i; \nm:=(a,b)->[g[2](a[1],b[1]),h[2](a[2],b[2])];\ni:=a- >[g[4](a[1]),h[4](a[2])]; \n[[seq(seq([g[1][i],h[1][j]],j=1..nops(h[1] )),i=1..nops(g[1]))], \nm, [g[3],h[3]], i] end: " }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 28 "For example, the product of " }{XPPEDIT 18 0 "Z[5] ;" "6#&%\"ZG6#\"\"&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Z[7];" "6#&% \"ZG6#\"\"(" }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A:=Z(5) &x Z(7):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Check i f it is an extended group:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "type(A, extendedGroup);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Find the order of it:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "nops(A[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#N" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "We ca n use it to find products of three or more groups, too:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B:=Z(2) &x Z(2) &x GL2(2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "type(B, extendedGroup);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "nops(B[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Here is the identity of it: " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$\"\"!F%7$7$\"\"\"F%7$F%F(" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 15 "It looks as if " }{TEXT 273 1 "B" }{TEXT -1 96 " equals the product of the first two groups multiplied by the third group. Ch eck if it is true: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "eval b( B[1] = ((Z(2) &x Z(2)) &x GL2(2))[1] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Is a \+ Group Cyclic?" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The following pro cedure is checking whether an extended group " }{TEXT 274 1 "G" } {TEXT -1 12 " is cyclic: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "IsCyclic:=proc(g) local v;\nv:=\{op(g[1])\}; while not nops(v)=0 \+ and not Ord(v[1],g)=nops(g[1]) do v:=v minus \{op(Cycle(v[1],g))\} od; not evalb(v=\{\}) end: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Here \+ are a few examples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "IsCy clic(Z(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "IsCyclic(Cyclic(25));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "IsCyclic(Dihedral(12));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&fal seG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "IsCyclic(Un(48));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "IsCyclic(Un(49));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "IsCyclic(Z(4) \+ &x Z(6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "IsCyclic(Z(4) &x Z(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "If a group is cyclic, the following procedure finds its generators : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "Generators:=proc(g) \+ local v,n; n:=nops(g[1]); if n=1 then g[3] else \nv:=\{op(g[1])\}; whi le not nops(v)=0 and not Ord(v[1],g)=n do v:=v minus \{op(Cycle(v[1],g ))\} od; if v=\{\} then FAIL else \{seq(Cycle(v[1],g)[un(n)[i]+1],i=1. .phi(n))\} fi fi end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For exam ple, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Generators(Un(49)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.\"\"$\"\"&\"#5\"#7\"#<\"#C\"#E \"#L\"#Q\"#S\"#X\"#Z" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Gen erators(Z(4) &x Z(5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*7$\"\"\"\" \"$7$F%\"\"%7$F&F%7$F&\"\"#7$F&F&7$F&F(7$F%F%7$F%F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Generators(Un(48));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "Norma lizers and Conjugacy Classes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 290 " \"Normalizer\" is an environment variable in Maple, so we should use a nother term for the normalizer of a subgroup. I chose to use \"normali zer\" even if it contradicts the agreement of using capital letters fo r commands related to the extended groups. First we have to define con jugate groups:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Conjugate :=(x,h,g)->map(y->g[2](g[2](x,y),g[4](x)),h): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "For example, the group " }{XPPEDIT 18 0 "x*H*x^`-1`; " "6#*(%\"xG\"\"\"%\"HGF%)F$%#-1GF%" }{TEXT -1 27 " conjugate to the s ubgroup " }{TEXT 275 1 "H" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "D[3];" " 6#&%\"DG6#\"\"$" }{TEXT -1 35 " generated by the 6th element, for " } {TEXT 276 1 "x" }{TEXT -1 8 " = 2, is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Conjugate(2,[1,6],Dihedral(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "N ow we define normalizers by the following procedure: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "normalizer:=(h,g)->select(x->evalb( \{op(Conjugate(x,h,g))\}=\{op(h)\}),g[1]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "For example, the normalizer of the cyclic subgroup of " } {TEXT 277 2 "GL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[3];" "6#&%\"ZG6# \"\"$" }{TEXT -1 15 ") generated by " }{XPPEDIT 18 0 "matrix([[1, 1], \+ [0, 1]]);" "6#-%'matrixG6#7$7$\"\"\"F(7$\"\"!F(" }{TEXT -1 3 " : " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "map(matrix,normalizer(Cycle( [[1,1],[0,1]],GL2(3)),GL2(3)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7.- %'matrixG6#7$7$\"\"\"\"\"!7$F*F)-F%6#7$F(7$F*\"\"#-F%6#7$7$F0F*F+-F%6# 7$F4F/-F%6#7$7$F)F)F+-F%6#7$F;F/-F%6#7$7$F0F)F+-F%6#7$FBF/-F%6#7$7$F)F 0F+-F%6#7$FIF/-F%6#7$7$F0F0F+-F%6#7$FPF/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Conjugacy classes can be defined as follows (also without the capitalization): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "c l:=(a,g)->\{seq(g[2](g[2](g[1][i],a),g[4](g[1][i])),i=1..nops(g[1]))\} :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "For example, the conjugacy c lass of the 6th element of " }{XPPEDIT 18 0 "D[3];" "6#&%\"DG6#\"\"$" }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cl(6,Dih edral(3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"%\"\"&\"\"'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Another example, the conjugacy cla ss of " }{XPPEDIT 18 0 "matrix([[1, 1], [0, 1]])" "6#-%'matrixG6#7$7$ \"\"\"F(7$\"\"!F(" }{TEXT -1 4 " in " }{TEXT 278 2 "GL" }{TEXT -1 4 "( 2, " }{XPPEDIT 18 0 "Z[3];" "6#&%\"ZG6#\"\"$" }{TEXT -1 2 "):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "map(matrix,cl([[1,1],[0,1]], GL2(3)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<*-%'matrixG6#7$7$\"\"#F) 7$\"\"\"\"\"!-F%6#7$7$F,F)7$F+F)-F%6#7$7$F,F+F(-F%6#7$F*7$F+F+-F%6#7$7 $F)F+7$F)F,-F%6#7$F*F=-F%6#7$F9F5-F%6#7$F1F5" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "Group of Quaternions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "All the groups of order less than 12 are either groups i n our list of extended groups, or their products, except the group of \+ quaternions. We will add the group of quaternions " }{XPPEDIT 18 0 "Q [8];" "6#&%\"QG6#\"\")" }{TEXT -1 171 " to our list. It is a non-Abeli an group of order 8 having 1 element of order 1 (the identity), 1 elem ent of order 2 (negative 1), and 6 elements of order 4 (plus or minus \+ " }{TEXT 279 7 "i, j, k" }{TEXT -1 51 "). There are only 2 non-Abelian groups of order 8, " }{XPPEDIT 18 0 "Q[8]" "6#&%\"QG6#\"\")" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "D[4];" "6#&%\"DG6#\"\"%" }{TEXT -1 21 ". \+ The dihedral group " }{XPPEDIT 18 0 "D[4]" "6#&%\"DG6#\"\"%" }{TEXT -1 379 " has only 2 elements of order 4, the rotations by 90 degrees \+ and 270 degrees. Its other elements have degrees either 1 (identity), \+ or 2 (the reflections and the rotation by 180 degrees). Thus, if we fi nd a non-Abelian group of order 8 having more than 2 elements of order 4, it will be the group of quaternions. Try to find it among the subg roups of the groups we already know. " }{TEXT 280 3 "GL2" }{TEXT -1 3 "(2," }{TEXT 286 1 " " }{XPPEDIT 18 0 "Z[2];" "6#&%\"ZG6#\"\"#" } {TEXT -1 26 ") is too small to contain " }{XPPEDIT 18 0 "Q[8]" "6#&%\" QG6#\"\")" }{TEXT -1 76 " as a subgroup, it has only 6 elements. Try t he next smallest groups in the " }{TEXT 282 2 "GL" }{TEXT -1 5 " and \+ " }{TEXT 283 2 "SL" }{TEXT -1 7 " series" }{TEXT 284 1 "," }{TEXT -1 1 " " }{TEXT 281 2 "SL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[3];" "6#& %\"ZG6#\"\"$" }{TEXT -1 39 "). First, find its elements of order 4:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Q:=Nordlist(4,SL2(3)):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "map(matrix,Q);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7(-%'matrixG6#7$7$\"\"!\"\"\"7$\"\"#F)-F%6#7$7$ F*F*7$F*F,-F%6#7$7$F,F*F0-F%6#7$7$F)F,7$F*F)-F%6#7$F17$F,F,-F%6#7$F>F5 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Exactly 6 elements. If the g roup generated by them has 8 elements, then it is the group of quatern ions. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "map(matrix,Gen(Q, SL2(3)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<*-%'matrixG6#7$7$\"\"\" \"\"!7$F*F)-F%6#7$7$\"\"#F*7$F*F0-F%6#7$7$F0F)7$F)F)-F%6#7$F67$F)F0-F% 6#7$F+F/-F%6#7$F:7$F0F0-F%6#7$F1F(-F%6#7$FAF5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "8 elements, so it is the group of quaternions. Let's m ake an extended group from it: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Q8inSL:=[[[[1,0],[0,1]],[[2,0],[0,2]],op(Q)],(a,b)->mm(a,b,3), [[1,0],[0,1]],a->invSL(a,3)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 " We also need a function converting the matrices, or lists, to the stan dard " }{TEXT 285 8 "i, j, k " }{TEXT -1 27 " expressions. Let's do it : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "quat([[1,0],[0,1]]): =1: quat([[2,0],[0,2]]):=-1: quat(Q[1]):=i: quat(Q[2]):=j: \nquat(Q[3] ):=-k: quat(Q[4]):=-i: quat(Q[5]):=k: quat(Q[6]):=-j:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "We will also need the backward conversion , from " }{TEXT 288 7 "i, j, k" }{TEXT -1 15 " to the lists: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "qback(1):=[[1,0],[0,1]]: qb ack(-1):=[[2,0],[0,2]]: qback(i):=Q[1]: qback(j):=Q[2]: \nqback(-k):=Q [3]: qback(-i):=Q[4]: qback(k):=Q[5]: qback(-j):=Q[6]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Now we can redefine the quaternion group \+ in terms of " }{TEXT 289 1 "i" }{TEXT -1 2 ", " }{TEXT 290 1 "j" } {TEXT -1 6 ", and " }{TEXT 291 1 "k" }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Q8:=[map(quat,Q8inSL[1]),(a,b)->quat(Q8i nSL[2](qback(a),qback(b))),1,a->quat(Q8inSL[4](qback(a)))]: " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For example, " }{MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Center(Q8) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "cl(i,Q8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #<$,$%\"iG!\"\"F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Cycle( i,Q8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"%\"iG!\"\",$F%F&" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Certainly, we could define the group of quaternions dire ctly from its Cayley table on p. 89 of Dr. Gallian's text. " }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Exercises" }}{EXCHG {PARA 14 "" 0 "" {TEXT 292 1 "1" }{TEXT -1 23 ". Find the subgroup of " }{TEXT 293 1 "U" }{TEXT -1 27 "(96) generated by 5 and 7. " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 294 1 "2" }{TEXT -1 23 ". Find the subgroup of " }{TEXT 295 2 "GL" }{TEXT -1 4 "(2, " }{XPPEDIT 18 0 "Z[4];" "6#&%\"ZG6 #\"\"%" }{TEXT -1 15 ") generated by " }{XPPEDIT 18 0 "matrix([[0, 1], [1, 2]]);" "6#-%'matrixG6#7$7$\"\"!\"\"\"7$F)\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "matrix([[1, 0], [0, 3]]);" "6#-%'matrixG6#7$7$\"\" \"\"\"!7$F)\"\"$" }{TEXT -1 3 " . " }}}}}}{MARK "1 6 18 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }